Papers

André planes of order $q^t$ can be constructed from the Desarguesian plane $PG(2,q^t)$ using a process that involves replacing André nets. This article constructs ovals in André planes of order $q^t$, $q$ even and $t$ prime that are inherited from conics of the Desarguesian plane $PG(2,q^t)$. Further, it is shown that certain conics do not inherit to arcs under the process of André replacement.

Let $S$ be a finite set of primes and write $\mathbb{Z}_S$ for the set of those non-zero integers whose prime divisors belong to $S$. Hajdu proved that the abc conjecture implies that the number of terms of any arithmetic progression in $H_S=\{\eta x^l\mid \eta\in \mathbb{Z}_S, x,l\in \mathbb{Z},\ \textnormal{with}\ x>0 \ \textnormal{and} \ l\geq 2 \}$ is bounded. Moreover, if $k\geq 3$ and the exponents of the powers are all $\geq 4$, then the number of such progressions are finite. We consider other sets and prove similar statements for these sets.

Beginning in the 1980s, Mukai introduced birational models of some moduli spaces of curves and some moduli spaces of K3 surfaces. They are defined as geometric invariant theory quotients. Little is known about the boundaries of these spaces. We describe an approach to efficiently evaluate certain invariant polynomials associated to these GIT problems. This allows us to show that several singular curves and surfaces are GIT semistable in Mukai's models. In the appendix, we give a combinatorial formula for an $\operatorname{SL}_n$-invariant in terms of the Gelfand-Tsetlin basis.

The small flow boundary property (SFBP), introduced by Burguet for fixed-point free topological flows, is a non-trivial generalization of the small boundary property (SBP). We characterize when a time-discretization of such a flow satisfies the SBP and deduce that an SFBP flow admitting an aperiodic time-discretization has vanishing mean dimension. Furthermore, we introduce a new quantity, flow-generated entropy, for a factor between a flow and a time-discretization, quantifying the dynamical complexity inherited from the flow itself. This is used in order to establish that any time-discretization of a flow with SFBP admits factors of arbitrarily small flow-generated entropy separating any fixed pair of distinct points. The argument relies on a construction of a dynamical version of the Cantor staircase function. Finally, the appendix includes proofs of fundamental properties of the marker property which have not yet appeared in the literature.

This article explores to what extent the geometry of gradient Ricci solitons extends to non-gradient Ricci solitons. The primary tool is the energy function $E$ of the soliton. We study consequences of various bounds on $E$. Under mild assumptions on the scalar curvature, we prove a weighted $L^1$-Liouville type theorem for both the usual Laplacian and the drifted Laplacian $\Delta_V$ associated to soliton vector field $V$, the former of which implies that Ricci solitons with bounded energy function have at most one nonparabolic end. Finally, we show that the measure $e^{-E} \mathrm{d} \mathrm{Vol}_{g}$ is finite for complete shrinking Ricci solitons, partially generalizing a result of Aaron Naber. As a consequence, non-gradient shrinking Ricci solitons also have finite fundamental groups, as in the gradient case.

Recovering structural information from noisy high-dimensional data is a fundamental task in statistical inference. We investigate the recovery thresholds for a graph hidden in a randomly weighted complete graph. Specifically, an unknown graph $H^* \in H_n$ is chosen uniformly at random, and hidden in a complete graph of $n$ vertices as follows: the weight of an edge $e \in H$ is distributed independently according to $P_n$; otherwise the weight is distributed independently according to $Q_n$. The goal is to recover almost all of $H$ from these edge weights. Assuming a local Lipschitzness of the Rényi divergence between distributions $P_n$ and $Q_n$, and a mild density condition for the graphs $H_n$, we give a unified characterization of the information-theoretic limit for recovering almost all of $H$ (also known as almost exact recovery). Our characterization connects the KL divergence between $P_n$ and $Q_n$ to the logarithm of the first moment threshold of $H$ in the Erdős-Rényi random graph model $G(n,p)$. Our lower bound also extends to the task of partial recovery, in which only a constant $\lambda$-fraction of $H$ needs to be recovered. Last but not least, for certain Bernoulli and Exponential regimes, and for Gaussian distributions, we are able to show an All-or-Nothing (AoN) threshold phenomenon at the exponential scale.

High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension. We propose Riemannian metric matching: a denoising probabilistic framework for learning the Riemannian geometry of data using neural networks. Specifically, we learn the carré du champ operator, which, using diffusion geometry, gives us access to the Riemannian geometry toolkit for downstream machine learning and statistical tasks. Our key observation is that the carré du champ operator can be formulated as a conditional expectation over random perturbations of the data, which can be exploited for sample-wise training and constant cost, amortized inference without explicit kernel construction. Empirically, metric matching rivals or improves the accuracy of $k$-NN-based diffusion geometry estimators, while enabling amortized inference that is up to $400\times$ faster, and supports graph-free geometric analysis on high-dimensional images where nearest neighbors break down.

It is well known that finite groups with exactly two character degrees have an abelian derived subgroup and, consequently, are solvable. Let $G$ be a finite group and $N$ a normal subgroup of $G$. In this paper, we prove that normal subgroups possessing exactly two degrees of minimal $G$-invariant characters are solvable. Furthermore, it is shown that if these degrees are $\{1, f\}$ for some integer $f$, then either $f$ is a prime power or the commutator subgroup $[N,G]$ is abelian. Whether $[N, G]$ is abelian when $f$ is a prime power remains an open problem. Specifically, we prove that this holds when $f=p$.

Simulating high-enthalpy fractured geothermal reservoirs is challenging due to the complex coupled processes of non-isothermal, multiphase, multicomponent flow, strongly nonlinear thermodynamics, and the dominant role of fractures. These complexities are amplified by mineral scaling, such as halite precipitation, which can impair reservoir permeability and well productivity. To address this, we present a new compositional flow model based on a persistent set of primary variables (pressure, enthalpy, and overall salt mass fraction). The formulation naturally handles phase transitions without manual switching, enhancing numerical stability. The model integrates a discrete fracture-matrix approach and employs an efficient, robust correlation-based phase-behaviour linearisation of saltwater thermodynamics, replacing expensive on-the-fly phase separation calculations. It incorporates the Kozeny-Carman relation to dynamically model porosity and permeability reduction from halite precipitation. Implemented in the open-source PorePy framework, the model is verified through a 1D salt dissolution benchmark against the established closed-source simulator CSMP++, showing strong agreement across geothermal conditions involving transitions between single- and multi-phase regions. Application to a 2D halite-saturated fractured reservoir with injection and production demonstrates the model's capability to predict halite precipitation patterns and their impact on permeability damage and energy recovery. Numerical results further show the model's value in predicting operational challenges such as wellbore blockage and the role of fracture connectivity. The model thus provides an open-source numerical tool for analysing complex heat and mass transport with mineral scaling in high-enthalpy fractured geothermal systems.

This paper investigates the growth of the sequences $(\|T^n(I-T)^k\|)_{n\ge 1}$ for bounded linear operators on Banach spaces under various resolvent conditions. Focusing on $\alpha$-Ritt operators that are also strongly Kreiss bounded, we show that the growth estimates established by Nevanlinna for Kreiss bounded operators can be significantly refined within the Hilbert space setting, nearly attaining the optimal rates obtained by Seifert for the more restrictive power-bounded case. We further introduce the class of $(\alpha, \beta)$-RK operators as a generalization of both Ritt and Kreiss-type conditions. For these operators, we derive comprehensive growth estimates which, for certain ranges of $\alpha$ and $\beta$, yield improvements over existing bounds in the literature. Particular attention is given to $\beta$-Kreiss operators, for which we provide a characterization via Cesàro-type means. We show that, in contrast to the well-known case $\beta = 1$, the power growth estimate $\|T^n\| = O(n^{\beta})$ is sharp whenever $\beta > 1$. The optimality of our estimates is discussed in several cases, relying on constructions and techniques developed by Nevanlinna, Spijker, and Borovykh. We conclude by providing a characterization of Ritt operators that appears to be absent from the literature.

We show that every Polish extended metric measure space arises as an inverse limit of metric measure spaces up to isomorphism. We then prove that synthetic Ricci curvature lower bounds and several functional inequalities, including the log-Sobolev, Talagrand, Poincaré, and dimension-free Harnack inequalities are stable under inverse limit. We discuss applications to infinite-dimensional spaces, including abstract Wiener spaces and their quotient spaces.

We completely characterize when three equal-order subgroups of a finite abelian group share a common transversal. Consequently, we determine when three full-rank lattices of equal volume with a discrete sum in $\mathbb{R}^d$ admit a common fundamental domain, answering, for $n=3$, an open question from 1997.

We derive an explicit formula for the sectional curvature of the space ${\cal M}(M)$ of finite measures on a Riemannian manifold M. The space ${\cal M}(M)$ is equipped with the Hellinger-Kantorovich metric $HK$. Even in the case M=R^n, the curvature is comprised of two parts: the `lifted part' is negative, and the `twisted part' is positive. It will be analyzed in detail for the multidimensional torus. Our general approach to sectional curvature in geodesic spaces also leads to new insights into the curvature of the space $P_2(M)$ of probability measures on M equipped with the Kantorovich-Wasserstein metric $W_2$.

Classical high-order backward differentiation formula (BDF) methods for the Landau-Lifshitz-Gilbert (LLG) equation often suffer from restrictive stability constraints, requiring small time steps and imposing stringent lower bounds on the damping parameter. These limitations become particularly severe for schemes of order higher than three. In this paper, we develop a class of high-order generalized BDF (GBDF) schemes for the LLG equation, including both semi-implicit and fully explicit treatments of the gyromagnetic term. The proposed schemes significantly improve stability properties and substantially relax the damping parameter constraints, but introduce essential difficulty in its analysis compared to the classical BDF schemes. We construct a novel multiplier which enables us to carry out a energy-based error analysis. This approach yields optimal-order error estimates under considerably weaker assumptions on the damping parameter than those required for classical BDF schemes. Numerical experiments are presented to confirm the theoretical results, and demonstrate that the proposed GBDF schemes achieve higher accuracy, enhanced stability, and much wider admissible damping regimes compared to classical high-order BDF methods.

We study residue restrictions for a two-color partition series $P(q)=(q^4;q^4)_\infty S(q)$ arising from work of Andrews and Bachraoui on partitions with odd smallest part. Motivated by the explicit exponent structure in the Bailey-transform formulas for the associated generating function, we obtain elementary restrictions on the support of the coefficients. In particular, we show that the residue class $4\pmod8$ does not occur, and we prove a further refinement modulo $16$, yielding several vanishing classes for the coefficients of the series. Our argument is completely residue-theoretic and avoids the use of modular completions.

In recent years, the technique of convex integration has demonstrated that for some initial data (also referred to as 'wild initial data'), many PDE models of mathematical fluid mechanics allow for a multitude of admissible (weak) solutions. This paper is concerned with the question regarding how large the set of such wild initial data is for the isentropic Euler equations. We prove that wild initial data form a dense set in $L^r$ for any $r \in [1,\infty)$. In contrast to existing results in the literature, in this paper 'wild initial data' are data which give rise to infinitely many global-in-time weak solutions which are admissible in the sense that the local energy inequality holds. In other words, the set of initial data with infinitely many admissible weak solutions (independent of the choice of time interval) is dense. A novel part of the construction is that we use a measure-valued (dissipative) solution as the ansatz for the subsolution. This requires several new ideas, in order to ensure the required regularity of the subsolution and to obtain a lower bound for the density. Another crucial ingredient of the proof is that the (local) energy density and the energy flux are constructed as part of the convex integration scheme, in order to obtain solutions which adhere to the local energy inequality.

Sliver elements are an intrinsic difficulty of three-dimensional tetrahedral mesh generation and remain costly, and sometimes impractical, to eliminate completely. Although isolated degenerate elements do not necessarily prevent finite element convergence, connected clusters or sheets of slivers may impose artificial constraints on the discrete solution, leading to locking and severe loss of accuracy. In this work, we revisit the effect of slivers from the viewpoint of the finite element solution and propose a robust solver-side treatment based on the Tempered Finite Element Method (TFEM). The method limits the singular contribution of degenerate elements by introducing a lower bound on the Jacobian determinant, which can be interpreted as a vanishing added-volume correction. The resulting formulation prevents the effective element volume from falling below a threshold while preserving the relevant physical modes of the solution. We analyze the stiffness matrices of degenerate tetrahedra, identify the mechanisms responsible for locking in sliver bands, and assess the method on a range of representative physical problems, including incompressible flow, Cahn–Hilliard phase-field dynamics, transient wave propagation, and vibro-acoustic fluid–structure interaction. The numerical results show that TFEM consistently recovers accurate and physically meaningful solutions on meshes for which standard FEM exhibits locking or loss of convergence, providing a simple and broadly applicable alternative to exhaustive geometric sliver removal.

This short paper has two goals. First, explaining a simple procedure (which is essentially folklore) that, sometimes, makes it possible to obtain a formula for the number of solutions to a system of multivariate polynomial inequalities over a finite field. Second, applying that procedure to prove a formula for the number of contiguous superregular $4 \times 4$ matrices over a finite field. The formula was previously conjectured by Appuswamy, Bazzani, Connelly, Ekaireb, Congero, and Zeger [Probability of super-regular matrices and MDS codes over finite fields, arXiv:2603.20983]. In addition, the same procedure is used to provide formulas for the number of contiguous superregular $3 \times 4$, $3 \times 5$, and $3 \times 6$ matrices over a finite field.

This paper presents a random approach to the multibonacci sequence. We generalise the model introduced by Benjamin, Levin, Mahlburg, and Quinn, which is based on a random tiling method using dominoes and squares that leads to the Fibonacci sequence, and which was extended to the tribonacci case in a previous work by the authors. Our approach employs tiling with linear $k$-ominoes, $k=1,\ldots,s$, combined with specific colouring, to generate a weighted multibonacci sequence. For a natural random variable~$X$ defined by this model, we establish the distribution of $X$ in terms of multibonacci numbers and compute $\mathbb{E}[X] = 2^{s+1}-3$.

Population-based optimizers combine update rules such as mutation, selection, and recombination. When one rule changes, it is often unclear which convergence guarantees survive or how the new combination should be assessed. We develop an operator calculus: an operator is a population-update rule, and the calculus specifies how separately checked effects can be combined. Under explicit regularity and small-step conditions, the leading changes caused by the updates add, yielding reusable building blocks for convergence analysis. The framework distinguishes finding and retaining a good solution, reducing the population's mean objective, and concentrating candidates near an optimizer, and identifies the extra approximation conditions needed for finite evaluation-budget guarantees. Applications include distribution adaptation, recombinative evolution, and consensus dynamics, with verified nonconvex cases. Controlled experiments on a common nonconvex problem collection show how component effects change with population geometry and the performance measure: a rule can worsen the mean objective yet produce better candidates.

Consider the Chevalley map $$ p:\mathfrak{gl} _n(F)\to (\mathfrak{gl}_n//\mathrm{GL}_n)(F), $$ where $F=\mathbb{F}_\ell((t))$. We show that the push forward via $p$ of every smooth compactly supported measure on $\mathfrak{gl}_n(F)$ is a measure whose density belongs to $L^q$ for every finite $q$. As a consequence, using the main result of [AGKSc], we obtain local integrability for Harish–Chandra's characters of irreducible cuspidal representations of $\mathrm{GL}_n(F)$.

We study the ways that one tiling space can be a finite regular cover of another. We classify all of the finite regular covers of a tiling space via its structure as an inverse limit space. If the tiling space $\Omega$ can be written as an inverse limit $\varprojlim \Gamma_n$, then the étale fundamental group of $\Omega$, which is defined via a limit of covers, is isomorphic to the inverse limit $\hat \pi_1(\Omega) := \varprojlim \hat \pi_1(\Gamma_n)$ of the profinite completions of the fundamental groups $\pi_1(\Gamma_n)$. This isomorphism allows us to construct all covers of tiling spaces and to use those covers to distinguish spaces that have identical cohomology groups.

Let $E$ be a non-isotrivial elliptic curve over a global function field $\mathbb{F}_q(t)$ of characteristic coprime to $2$ and $3$. Under some explicit conditions, we determine the secondary terms for the first moments of prime Selmer groups of cyclic prime twist families of $E$ over $\mathbb{F}_q(t)$.

In this paper, we construct nonconforming finite element spaces $\boldsymbol{V}^{\mathbf{d}\cap\mathring{\boldsymbol{\delta}}}_h\Lambda^k$ for the approximation of $H\Lambda^k\cap H^*_0\Lambda^k$ on simplicial meshes, for $n\ge 2$ and $1\le k\le n-1$, by enforcing adjoint continuity against piecewise Whitney spaces rather than trace matching. It holds, with $\mathbf{d}^k_h$ and $\boldsymbol{\delta}_{k,h}$ denoting respectively the piecewise action of differential and codifferential operators, and $\boldsymbol{\mathfrak{H}}_h\Lambda^k$ being the discrete harmonic forms in the FEEC sense, that $\boldsymbol{\mathfrak{H}}_h\Lambda^k=\{\boldsymbol{\mu}_h\in \boldsymbol{V}^{\mathbf{d}\cap\mathring{\boldsymbol{\delta}}}_h\Lambda^k:\mathbf{d}^k_h\boldsymbol{\mu}_h=0,\ \boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h=0\}$, which mirrors the continuous Hodge–Laplace kernel on domains with nontrivial topology. The space is not a classical Ciarlet-type finite element space; though, a uniform discrete Poincare inequality and locally supported basis functions (supported on at most two cells) are guaranteed. The resulting primal scheme yields an $O(h)$ error bound for smooth data and $O(h^s)$ on $s$-regular domains ($0<s\le 1$), nontrivial topology admitted. Two- and three-dimensional eigenvalue tests agree with the mixed method on perforated domains, which are given to verify the validity of the scheme.

We address the existence of normal forms for rough ordinary differential equations. We assume suitable smoothness and the hyperbolicity of an equilibrium point. In this context, we establish local formal equivalence of the two solution flows generated by a random nonlinear RDE and its linearized version. This provides the foundation for extending normal form theory to rough differential equations.

We propose a median digital-net algorithm for $L^2$-approximation of non-periodic functions over $[0,1]^s$, inspired by the recently developed median lattice algorithms for the periodic setting. The algorithm requires no smoothness or weight parameters but only a sufficiently large candidate Walsh index set $K$. It proceeds in three stages: generating multiple estimates of the Walsh coefficients in $K$ using independent randomized digital-net samples; taking the respective median of both the estimates and their absolute values; then, based on these median values, identifying the dominant coefficients and constructing a truncated Walsh series as the final approximation. We prove that if the target function has dominating mixed partial derivatives up to order $\alpha$, all having finite Vitali variation of fractional order $\lambda$, then the algorithm achieves an $L^2$-error of $\mathcal{O}(M^{-\alpha-\lambda+\eta})$ with high probability, where $M$ is the total number of function evaluations and $\eta>0$ is arbitrarily small. Furthermore, the implied constant grows at most polynomially in the dimension $s$ under suitable decay conditions on the ANOVA components of the target function. On the implementation side, we provide both parameter-dependent and -independent constructions of the index set $K$, and employ the fast Walsh–Hadamard transform and Gray code ordering to accelerate the algorithm. Numerical experiments support the theoretical analysis and demonstrate that the proposed algorithm remains effective in high-dimensional settings.

We construct quantum dynamical Weyl group elements associated with quantum loop groups of arbitrary shuffle type. Using the construction, we define the quantum lattice operator and the algebraic quantum difference equations for each tensor products of semisimple modules $V$ in category $\mathcal{O}$. We prove that algebraic quantum difference operators form a family of commuting operators, and they also commute with the qKZ operators for the tensor products of modules of the above type in $\mathcal{O}$. This recovers the construction in \cite{OS22} and can be viewed as the difference analog of the trigonmetric Casimir connection when the quantum loop group corresponds to the finite type symmetrisable Cartan matrix.

In this article we propose a novel thermodynamical model which couples the evolution of the thickness of a grounded shallow ice sheet with the evolution of its internal temperature. Both the grounded shallow ice sheet surface elevation and the ice internal temperature are subjected to physical constraints. The equations governing the evolution of the grounded shallow ice sheet thickness are degenerate, and the ice internal temperature evolves in a moving domain. First, we formally model the phenomenon under consideration by adopting strategies akin to those employed in the construction of diffuse-interface models. Second, we establish the existence of solutions for one such formal model by means of the penalty method, and we observe that the low regularity of the problem under consideration prevents us from obtaining a standard concept of solution.

We study the time-asymptotic stability of viscous-dispersive shock waves for the outflow problem of the barotropic Navier–Stokes–Korteweg equations, which describe viscous fluids with internal capillarity. Assuming that the far-field state is subsonic or transonic and that the velocity at the boundary is larger than the far-field velocity, we prove that the solution converges to the corresponding viscous-dispersive shock wave as $t \to +\infty$, provided that the shock amplitude and the initial perturbation are sufficiently small. The proof is based on the method of $a$-contraction with shifts (for viscous equations) introduced in \cite{KV17,KV21,KVW23}. A main difficulty comes from controlling the boundary effect of the viscous-dispersive shock wave, as well as the influence of capillarity near the boundary.

The asymmetric simple exclusion process (ASEP) is an important particle model with deep connections to orthogonal polynomials. Motivated by this connection, Corteel and Williams introduced the Koornwinder moments $M^{Z}_{\lambda}$ at $ t=q $, which generalize the moments of Askey–Wilson polynomials. They showed that the partition function of the two-species ASEP is equal to $M^{Z}_{\lambda}$ for a one-row partition $ \lambda $. In this paper, we investigate a conjecture of Rains on the positivity of the minimal numerator of the Koornwinder moment $M^{Z}_{\lambda}$. We derive the first explicit formula for this moment, thereby obtaining a precise formulation of the conjecture by determining the minimal denominator of $M^{Z}_{\lambda}$. We also propose a generalization of the conjecture for the more general Koornwinder moments $M^{Z}_{\lambda,\mu}$ indexed by two partitions at special parameter values. We prove the generalized Rains' conjecture in two special cases: $(\xi,q)=(1,0)$ and $(\xi,q)=(1,1)$. For $(\xi,q)=(1,0)$, we construct a lattice path model and obtain a combinatorial formula for $M^{Z}_{\lambda,\mu}$ in terms of non-intersecting lattice paths. For $(\xi,q)=(1,1)$, we establish an explicit product formula for $M^{Z}_{\lambda,\mu}$ and give a combinatorial interpretation using lecture hall tableaux.